In an earlier paper (B. J.
Gardner, Pacific J. Math., 33 (1970), 109116) the torsion classes of abelian groups
which are closed under pure subgroups were characterized, and §§36 of the
present paper are devoted to generalizations of results appearing there. If
𝒞 is a homomorphically closed class of objects in an abelian category, a
subobject A of an object B is called 𝒞pure if it is a direct summand of every
intermediate subobject X for which X∕A ∈𝒞. (This terminology is due to C. P.
Walker). In particular, 𝒞 may be a torsion class. The following question is
investigated: If 𝒯^{−} and 𝒰 are torsion classes of abelian groups, when is 𝒯^{−} closed
under 𝒰pure subgroups? Although ordinary purity is not 𝒰purity for any
torsion class 𝒰, a torsion class 𝒯 is closed under pure subgroups if and only if
it is closed under 𝒯_{0}pure subgroups, where 𝒯_{0} is the class of all torsion
groups.
In §5, for an arbitrary torsion theory (𝒰,𝒢) a rank function ( 𝒰rank) is defined
for nonzero groups in 𝒢. It is: shown that every torsion class closed under 𝒰pure
subgroups. is determined by its intersection with 𝒰 and the groups of 𝒰rank 1 it
contains. When 𝒰 = 𝒯_{0}, the groups with 𝒰rank 1 are the rational groups, so the
earlier results for ordinary purity suggest that in general some refinement of the
representation should be possible.
A further special case of the general problem is als α solved: Let X and Y be
rational groups, T(X),T(Y ) the smallest torsion classes containing them. If X is a
subring of the rationals then T(X) is always closed under T(Y )pure subgroups;
if not, the condition is satisfied if and only if X has a greater type than
Y.
§7 is devoted to proving the following result: A torsion class is closed under
countable direct products, i.e. direct products of countable sets of groups, if and only
if it is determined by torsionfree groups.
